Hi! I need help to solve a task.
Write an equation of hyperbola. Foci coincides with the ellipse (x^2/35+y^2/10=1) focal points. There is given also a point from hyperbola - L(5sqrt(2);4)
Thanks for help in advance.
1. With an ellipse the foci have the coordinates $\displaystyle F_1(-e,0)~,~F_2(e,0)$ with $\displaystyle e = \sqrt{a^2-b^2}$
That means the foci are at $\displaystyle F_1(-5,0)$ and $\displaystyle F_2(5,0)$
2. A hyperbola is defined by a constant difference of the distances between a point L on the hyperbola and the foci:
$\displaystyle |LF_1|-|LF_2| = 2a$
with $\displaystyle L(5\sqrt{2},4)$
I've got $\displaystyle a = \frac12 \cdot \sqrt{182-2\sqrt{3281}} $
3. The excentricity of a hyperbola is calculated by:
$\displaystyle e = \sqrt{a^2+b^2}$
Since the excentricity is e = 5 you can determine the value of b.
I've got $\displaystyle b=\frac12 \cdot \sqrt{2\sqrt{3281}-82}$
(For this part I used my computer)
4. Therefore the equation of the hyperbola is:
$\displaystyle \frac{x^2}{\left( \frac12 \cdot \sqrt{182-2\sqrt{3281}} \right)^2}-\frac{y^2}{\left( \frac12 \cdot \sqrt{2\sqrt{3281}-82} \right)^2}=1$
5. I've attached a drawing of the ellipse and the hyperbola with the point L on the right branch.